Calculation of physical properties of atoms and ions based on analytical perturbation theory

Abstract

Analytical perturbation theory is adapted to calculate properties of multi-electron atoms and ions. Results from zero-and first-order approximations are presented analytically and have an accuracy comparable to that of the Hartree-Fock method. It is shown for the first time that the summation over two-electron intermediate states in calculations of correlation corrections can be done in closed form.Introduction. The history of methods for solving the Schroedinger equation (SE) for a system of N nonrelativistic electrons in the field of a nucleus with charge Z is almost as long as that of quantum mechanics itself. Nevertheless, calculation of the whole spectrum of states of such a system with N ≥ 2 to any required accuracy remains an open question despite the increase in computing power. Results of recent studies [1] attest to its urgency. They focused on development of methods for direct numerical solution of the SE for the ground state of the He atom. There currently exists a large number of approximation methods that satisfy mainly the demands of spectroscopy, condensedstate physics, or quantum chemistry. However, the problem of calculating accurately spectra of multi-electron systems is still important in principle. Approximation methods for solving the SE for such systems include the statistical Thomas-Fermi model (TFM) [2-5] and density-functional theory based on it, the Hartree-Fock method (HFM) [6], and other variational approaches. In particular, the energy of the ground and low-lying excited states can be calculated to very high accuracy for two-electron atoms and ions by selecting a trial wave function with a sufficient number of parameters [7,8]. However, the ability to generalize this approach to multi-electron systems seems problematical.The regular method for solving the SE for atoms and ions is perturbation theory, which is based on an expansion over powers of Z -1 [9]. However, its zero-order approximation is insufficiently accurate because it does not at all take into account electron coupling. A problem arises in calculating the subsequent corrections that is related to computation of sums of the two-electron coupling operator matrix elements over the spectrum of intermediate states.This can be done only approximately and for rather simple systems using the variational principle.Herein analytical perturbation theory (APT) is developed. It can be used to overcome largely these difficulties. The zero-order approximation of APT enables the properties of atoms and ions to be calculated with accuracy comparable to that of the HFM. Moreover, the expressions for the subsequent corrections can be written in a closed form that is suitable for calculations using existing numerical methods.Model Hamiltonian of Zero-Order APT. The initial Hamiltonian of the examined system has the form (Coulombic units are used [10]):